Birefringent lens interferometer for use in microscopy and other applications

ABSTRACT

Techniques to improve image quality in holography utilizing lenses made from materials with non-quantized anisotropic electromagnetic properties, such as birefringent materials, to advantageously split an incoming beam of light into two coincident beams with different focal lengths that interfere with one another and thus create holograms free of electro-optical or pixelated devices are disclosed for microscopy and other applications. The use of thin birefringent lenses and single crystal alpha-BBO lenses are introduced. Corresponding systems, methods and apparatuses are described.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.15/588,096 filed May 5, 2017, which claims the benefit of priority toU.S. Provisional Application Ser. No. 62/332,857 filed on May 6, 2016,and also claims the benefit of priority to and is a continuation-in-partof U.S. patent application Ser. No. 15/455,863 filed on Mar. 10, 2017now abandoned. U.S. patent application Ser. No. 15/455,863 filed on Mar.10, 2017 claims the benefit of priority to U.S. Provisional ApplicationSer. No. 62/306,537 filed on Mar. 10, 2016, and also is acontinuation-in-part of U.S. patent application Ser. No. 15/228,386filed on Aug. 4, 2016 now U.S. Pat. No. 10,289,070 dated May 14, 2019.U.S. patent application Ser. No. 15/228,386 claims the benefit ofpriority to U.S. Provisional Application Ser. No. 62/202,655 filed onAug. 7, 2015, and is a continuation-in-part of and claims the benefit ofPCT Application Serial Number PCT/US2015/028477 filed on Apr. 30, 2015,which claims the benefit of U.S. Provisional Application Ser. No.61/987,205, filed on May 1, 2014. The entire contents of U.S.Provisional Application Ser. No. 62/332,857, U.S. patent applicationSer. No. 15/455,863, U.S. Provisional Application Ser. No. 62/306,537,U.S. patent application Ser. No. 15/228,386, U.S. ProvisionalApplication Ser. No. 62/202,655, PCT Application Serial NumberPCT/US2015/028477 and U.S. Provisional Application Ser. No. 61/987,205are herein incorporated by reference.

GOVERNMENT RIGHTS

This invention was made with U.S. government support under grantR44CA192299 awarded by the National Cancer Institute (NCI). The U.S.government has certain rights in the invention.

FIELD

This disclosure relates to collecting and/or using Fresnel IncoherentCorrelation Holography (FINCH) or other holography images generated byuse of a birefringent lens or optical element to alter the phaseproperties of the received light or other electromagnetic radiation.

BACKGROUND

Holograms are records of the interference patterns created by two ormore light or other radiation waves. In order for the waves to interferethey must have different phase properties. In current holography methodsthe waves that are to be interfered are passed through different opticalpaths that impart different phase properties on each wave. In one classof methods of single-path holography, the waves are commonly givendifferent phase properties by being passed through or reflected off ofdigitized phase patterns displayed on a spatial light modulator (SLM) orother optical element. In another class of methods for self-interferenceholography, the waves originate from a single wave and are split by abeam splitter, then reflected off differing mirrors before beingrecombined in the last part of the beam path and brought to interfere.All of these methods produce holograms that may suffer from significantdefects due to slight mismatches in optical path length, quantizationerrors or undesired diffraction effects of the SLM or other opticalelement. Apparatuses, systems and/or methods that allow all the waves topass in the same optical path while receiving different phaseproperties, without being subject to unnecessary reflections orquantization errors or undesired diffraction effects, would be desirablein the field of holography.

SUMMARY OF EXAMPLE EMBODIMENTS OF THE INVENTION

Accordingly, one object of example embodiments is to provide anapparatus with non-quantized anisotropic electromagnetic properties usedto create electromagnetic interference from received electromagneticradiation, and a method for its use. The anisotropic electromagneticproperties may derive from one or more anisotropic components such as athin birefringent lens, and may be further adjusted by combination withother materials. The received electromagnetic radiation may be fromsources such as x-rays, black body radiation, infrared light, or lightof any wavelength from any source, coherent or incoherent. In someembodiments, the received electromagnetic radiation may be from amicroscope specimen and/or from a microscope. In the apparatus, thereceived electromagnetic radiation is then transformed by refractionand/or diffraction into two or more differentially modulated wavespropagating in a common path, and the modulated electromagnetic wavescreate the electromagnetic interference, which can take the form of aFresnel, Fourier, Fresnel Incoherent Correlation Holography (FINCH),off-axis or other hologram. The interference is recorded by a recordingdevice, and information about the source of the received radiation canbe obtained from the interference.

Another object of example embodiments is to provide an apparatus withnon-quantized anisotropic electromagnetic properties used to createelectromagnetic interference from received electromagnetic radiation,and a method for its use. The anisotropic electromagnetic properties mayderive from one or more anisotropic components such as a thinbirefringent lens, and may be further adjusted by combination with othermaterials. The received electromagnetic radiation may be from sourcessuch as x-rays, black body radiation, or light of any wavelength fromany source, coherent or incoherent. In some embodiments, the receivedelectromagnetic radiation may be from a microscope specimen and/or froma microscope. In the apparatus, the received electromagnetic radiationis then transformed by refraction and/or diffraction into two or moredifferentially modulated waves propagating in a common path withprogrammed differences between the modulations. The modulatedelectromagnetic waves create the electromagnetic interference, which cantake the form of a Fresnel, Fourier, FINCH, off-axis or other hologram.The interference is then used to deliver the programmed information to asubsequent device or object such as a microscope sample or opticalrecording medium.

Another object of example embodiments is to provide the advantageslisted above in configurations that do not require external powersources, allowing interference waves (and holograms) to be obtained in aportable manner.

An example embodiment provides an apparatus with non-quantizedanisotropic electromagnetic properties configured to createelectromagnetic interference from received electromagnetic radiation.The anisotropic electromagnetic properties of the apparatus may existindependent of external power. The received electromagnetic radiation istransformed by refraction and/or diffraction using at least one thinbirefringent lens into two or more differentially modulated wavespropagating in a common path such that the modulated electromagneticwaves create the electromagnetic interference. The receivedelectromagnetic radiation may be, for example, fluorescent light,chemiluminescent light, bioluminescent light, infrared light, incoherentlight, coherent light, other type of light, x-ray or black bodyradiation. The anisotropic properties of the apparatus may be derived,for example, from calcite, alpha barium borate, beta barium borate (BBO)or other birefringent materials. In some implementations the anisotropicproperties may be derived from liquid crystal material. For example, theliquid crystal material encased in flat or positively or negativelycurved non birefringent materials, or may be encased in flat orpositively or negatively curved birefringent materials.

The electromagnetic interference created by the apparatus of the exampleembodiment may be a Fresnel hologram, a Fourier hologram, a FINCHhologram, or an off axis hologram, or other hologram. The receivedelectromagnetic radiation may originate from a microscope and/ormicroscope specimen, or from a DNA sequencing gel or system. Theelectromagnetic interference that is created maybe recorded, forexample, by an image recording device, or by a point source detector.The electromagnetic interference may be used as the excitation patternin scanning holography, used in an excitation source in a StructuredIllumination (SIM) imaging system, or may be used to record data in aholographic storage medium. The received electromagnetic radiation maybe coherent or incoherent and may originate from the readout of aholographic data storage medium or any combination of the previousmethods. The electromagnetic interference may be interpreted to recoverdata stored in a holographic storage medium.

The anisotropic electromagnetic properties of the apparatus of theexample embodiment may be contained in one or more birefringent lenses.The apparatus may be configured to allow any difference in focal lengthbetween the ordinary and extraordinary focal lengths of the combinedlens system to be achieved based on choices of the radii of curvaturefor each surface of the birefringent lens and the focal lengths of anyassociated standard (also referred to as classical) lenses. Some or allof the radii of curvature of the birefringent lens elements may beinfinity. In some implementations, the described lenses are combined inone unit, where the combination means is an optically transmittingsubstance such as, for example, air or optical cement.

The apparatus of the example embodiment may be configured such that thedispersive properties of the birefringent materials are used to create amultitude of spatially separated wavelength dependent holograms from abroadband electromagnetic radiation source. In such a configuration thespatially separated holograms are directed to separate areas forrecording or further use or modification by means of color filters ordispersive prismatic or grating elements. In some implementations, thesource of the received electromagnetic radiation may be a human eyeFundus, and the refracted electromagnetic interference may be recordedon a digital camera. In some implementations the source of the receivedelectromagnetic radiation may be a microscope objective lens, and therefracted electromagnetic interference may be used to create classicallyresolved or optically super-resolved images. In some implementations,other optical devices may be configured to alter the electromagneticinterference to achieve desired spatial, chromatic and temporalcharacteristics.

Another example embodiment provides a birefringent optical deviceconfigured to simultaneously create, from a single source, focused spotsat two or more different planes. The focused spots may be used asexcitation light in a microscope and are simultaneously focused upon twoor more object planes. The birefringent optical device may be amicroscope objective. In some implementations, the birefringent opticaldevice may be contained within the microscope objective lens, and may beused to focus laser excitation light into the sample.

Another example embodiment provides a non-quantized birefringent opticaldevice for creating Fresnel, FINCH, Fourier or other holograms fromreceived electromagnetic radiation. The example non-quantizedbirefringent optical device includes hybrid lenses of birefringentlenses that are created by the combination of birefringent andnon-birefringent materials to create polarization sensitive lenses withtwo or more focal lengths of any specification.

Another example embodiment provides a non-quantized birefringent opticaldevice configured to have any two different focal lengths by combinationof lenses of different birefringent materials. The example opticaldevice may be used to create holograms, such as, for example, Fresnel,FINCH, Fourier or other holograms from received electromagneticradiation. The spacing between the independent focal planes of thelenses (spacing factor) may be varied. The hybrid lenses of birefringentlenses may be created by the combination of birefringent andnon-birefringent materials to form polarization sensitive lenses withtwo or more focal lengths of any specification. In some implementationsof the example birefringent optical device, the birefringent opticaldevice may be contained within a microscope objective lens.

Another example embodiment provides a method to create electromagneticinterference from received electromagnetic radiation by using an opticaldevice such as a thin birefringent lens with non-quantized anisotropicelectromagnetic properties. The example method includes transforming thereceived electromagnetic radiation by refraction and/or diffraction intotwo or more differentially modulated waves propagating in a common path,and creating the electromagnetic interference using the modulatedelectromagnetic waves. The received electromagnetic radiation may be,for example, fluorescent light, chemiluminescent light, bioluminescentlight, incoherent light, coherent light, infrared light, other type oflight, x-ray, or black body radiation. The anisotropic properties may bederived from calcite materials, from alpha or beta barium boratematerials, or from any material that is anisotropic. In someimplementations the anisotropic properties may be derived from liquidcrystal material. For example, the liquid crystal material encased inflat or positively or negatively curved non birefringent materials, ormay be encased in flat or positively or negatively curved birefringentmaterials. The created electromagnetic interference may be a hologramsuch as, for example, a Fresnel hologram, a Fourier hologram, a FINCHhologram, or an off axis hologram. The received electromagneticradiation may originate from a microscope and/or microscope specimen, orfrom a DNA sequencing gel or system or any other object that emits orreflects light. The electromagnetic interference that is created may berecorded by an image recording device, or by a point source detector. Insome implementations the electromagnetic interference is used as theexcitation pattern in scanning holography, as an excitation source in aStructured Illumination (SIM) imaging system, or to record data in aholographic storage medium. In some implementations the receivedelectromagnetic radiation originates from the readout of a holographicdata storage medium. The electromagnetic interference may be interpretedto recover data stored in a holographic storage medium.

The example method may operate to use the dispersive properties of thebirefringent materials to create a multitude of spatially separatedwavelength dependent holograms from a broadband electromagneticradiation source. In some implementations, the source of the receivedelectromagnetic radiation may be a human eye Fundus, and the refractedelectromagnetic interference is recorded on a digital camera. In someimplementations, the source may be a microscope objective lens, and therefracted electromagnetic interference is used to create opticallysuper-resolved images

Another example embodiment provides a method for simultaneouslycreating, from a single source, focused spots at two or more differentplanes using a birefringent optical device. The focused spots may beused as excitation light in a microscope and are simultaneously focusedupon two or more object planes. The birefringent lens may be amicroscope objective. In some implementations, the birefringent opticaldevice may be contained within the microscope objective lens, and may beused to focus laser excitation light into the sample.

In some implementations, the example method may allow any difference infocal length between the ordinary and extraordinary focal lengths of thecombined lens system to be achieved based on choices of the radii ofcurvature for each surface of the birefringent lens and the focallengths of any associated standard lenses. Some or all of the radii ofcurvature of the birefringent elements may be infinity. In someimplementations, the described lenses may be combined in one unit, withan optically transmitting substance such as air or optical cement as thecombination medium.

Another embodiment provides a method of using non-quantized birefringentoptical devices with any two different focal lengths by combination oflenses of different birefringent materials. The method may be used tocreate holograms such as, for example, Fresnel, FINCH, Fourier or otherholograms, from received electromagnetic radiation. The differencebetween the focal lengths of the lenses may be varied. Hybrid lenses ofbirefringent lenses may be created by the combination of birefringentand non-birefringent materials to create polarization sensitive lenseswith two or more focal lengths of any specification.

Another example embodiment provides a method to use birefringent opticaldevices incorporating one or more birefringent spherical lenses to formlenses with two or more polarization sensitive focal lengths of anyspecification.

Another example embodiment provides a birefringent optical deviceincorporating one or more birefringent spherical lenses to obtain lenseswith two or more polarization sensitive focal lengths of anyspecification.

Another embodiment provides a birefringent device configured to createFresnel, FINCH, Fourier or other holograms from electromagneticradiation. The electromagnetic radiation may be light. The birefringentdevice is composed of a material that is birefringent at opticalwavelengths. The birefringent device may be used in conjunction withother optical devices to alter the hologram to achieve desired spatial,chromatic and temporal characteristics. The light beam that is processedby the birefringent device may originate from a microscope specimen. Thehologram that is created may be recorded by an image recording device.The light beam originating from the specimen may be fluorescent lightwhose emission was induced by standard microscopy methods. The lightbeam originating from the specimen may include fluorescent light whoseemission was induced and transmitted in a confocal arrangement, whoseemission was induced by multiphoton excitation, or whose emission wasinduced by nonlinear-optical methods. In some implementations, the lightbeam originating from the specimen is chemiluminescence light,transmitted light or reflected light. In some embodiments, the lightbeam that is processed by the birefringent optical device originatesfrom a camera lens, or from a biological sequencing gel. In someembodiments, the electromagnetic radiation is laser light.

In some example embodiments, the birefringent device which is configuredto create Fresnel, FINCH, Fourier or other holograms fromelectromagnetic radiation, which is composed of a material that isbirefringent at optical wavelengths, and which may be used inconjunction with other optical devices to alter the hologram to achievedesired spatial, chromatic and temporal characteristics, may becontained within a microscope objective lens. The light beam that isprocessed by the birefringent device may originate from a microscopespecimen. The hologram that is created may be recorded by an imagerecording device. The birefringent device within the microscopeobjective lens may be used to focus laser excitation light into thespecimen.

Another example embodiment provides a birefringent device configured tocreate Fresnel, FINCH, Fourier or other holograms from electromagneticradiation, in which the electromagnetic radiation may be light. Thebirefringent device is composed of a material that is birefringent atoptical wavelengths. The birefringent device may be used in conjunctionwith other optical devices to alter the hologram to achieve desiredspatial, chromatic and temporal characteristics for any given usagemodality. The hologram created by the birefringent device may be used asthe excitation pattern in scanning holography, providing significantincreases in stability over current methods. While scanning holographycurrently produces the Fresnel hologram used for excitation from a laserbeam passed through a modified Michaelson interferometer with two beampaths, some example embodiments us a single beam path through thebirefringent device. The single beam path avoids the problems ofdiffering properties of different beam paths, such as relativedifferences in vibration that can degrade the excitation pattern inconventional scanning holography. In some implementations of the examplebirefringent device configured to create Fresnel, FINCH, Fourier orother holograms, the hologram may be used to modulate the excitationbeam in a Structured Illumination (SIM) imaging system. For example, thebirefringent device may be used to impart linear phase difference in theSIM excitation beam instead of spherical phase difference; or abirefringent device with an axicon phase profile may be used; or theouter part of a Fresnel hologram formed form the excitation laser beam,that approximates a linear fringe pattern, may be used.

In some implementations of the example birefringent device configured tobe used in conjunction with other optical devices to alter the hologramto achieve desired spatial, chromatic and temporal characteristics, thehologram is used to record data in a holographic storage medium.

In some implementations of the example birefringent device configured tobe used in conjunction with other optical devices to alter the hologramto achieve desired spatial, chromatic and temporal characteristics, thelight creating the hologram originates from the readout of a holographicdata storage medium. The hologram may be interpreted to recover datastored in a holographic storage medium.

In some implementations of the example birefringent device, thebirefringent device is configured to use the dispersive properties ofthe birefringent materials create a multitude of spatially separatedwavelength dependent holograms from a broadband electromagneticradiation source. The electromagnetic radiation may be coherent,incoherent, fluorescent light, chemiluminescent light, light from amicroscope, or light from a DNA sequencing means.

Another example embodiment provides a birefringent optical deviceconfigured to focus excitation light into two object planes in a singleexposure. The birefringent lens may be a microscope objective.

Another example embodiment provides a birefringent optical device forcreating focused images of two differing object planes in a singleexposure. The birefringent lens may be a microscope objective.

Another example embodiment provides a birefringent optical devices tocreate Fresnel, FINCH, Fourier or other holograms from electromagneticradiation wherein hybrid lenses of birefringent lenses are created bythe combination of birefringent and non-birefringent materials to createpolarization sensitive lenses with two or more focal lengths of anyspecification.

Another example embodiment provides a method for holography wherein thechoices of the radii of curvature for each surface of the birefringentlens and the focal length of the associated standard lens allow anydifference in focal length between the ordinary and extraordinary focallengths of the combined lens system to be achieved. The described lensesmay be combined in one unit. The combining of the lenses may be by meansof an optically transmitting substance such as air and/or opticalcement.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1. A diagram depicting a conventional imaging lens wherein thereceived electromagnetic (EM) radiation from the object is focused toonly one plane of focus.

FIG. 2. A diagram depicting three configurations for Fresnel IncoherentCorrelation Holography (FINCH) imaging using a spatial light modulator(SLM) to produce the reference and sample beams.

FIG. 3. Schematic of a FINCH fluorescence microscope using Thin LiquidCrystal Gradient Refractive Index (TLCGRIN) lens.

FIG. 4. A birefringent lens with two focal lengths f₁ and f₂, accordingto one or more example embodiments.

FIG. 5. A generalized scheme for creating a FINCH hologram according toone or more example embodiments.

FIG. 6. The differing focal lengths of a birefringent lens resultingfrom the differing refractive indices in the transverse plane of thelens, according to one or more example embodiments.

FIG. 7. Wavelength dependent shift in location of optimal hologramplanes, according to one or more example embodiments.

FIG. 8. Point hologram raw and processed images captured from a laser asthe EM radiation source, using a FINCH system as in FIG. 3, according toone or more example embodiments.

FIG. 9a . Point hologram raw and processed images captured from a laseras the EM radiation source, using a FINCH system incorporating a calciteBRL, according to one or more example embodiments.

FIG. 9b . Comparative classical fluorescence microscopy and FINCHfluorescence microscopy of a standard object using a FINCH systemincorporating a calcite BRL, according to one or more exampleembodiments, demonstrating improved image contrast and resolution in theFINCH image.

FIG. 9c . Comparative classical fluorescence microscopy and FINCHfluorescence microscopy of standard sub-resolution bead objects using aFINCH system incorporating a calcite BRL, according to one or moreexample embodiments, demonstrating improved image resolution in theFINCH image with a comparative plot of the widths of the bead intensityprofiles measured by each method.

FIG. 9d . Comparative classical fluorescence microscopy and FINCHfluorescence microscopy of 110 nm bead objects using a FINCH systemincorporating a single crystal α-BRL, according to one or more exampleembodiments, demonstrating improved image resolution in the FINCH imagewith a comparative plot of the widths of the bead intensity profilesmeasured by each method.

FIG. 10. A schematic of two birefringent lenses used in tandem,according to one or more example embodiments.

FIG. 11. A schematic of a birefringent lens used in conjunction with aflat birefringent plate, according to one or more example embodiments.

FIG. 12. A schematic of a birefringent plate or block used to create twofocal planes from a single spherical glass lens, according to one ormore example embodiments.

FIG. 13. Arrangements of optics in a birefringent lens incoherentinterferometer in accordance with one or more example embodiments.

FIG. 14. Schematic illustration of a FINCH microscope according to someexample embodiments

DETAILED DESCRIPTION

In classical optical imaging, a beam of light is emitted or reflectedfrom an object, and is then collected by a lens. In the simplest case,the light beam is focused by this lens to create an image at a focalplane. The image is two-dimensional as shown in FIG. 1 depicting a lens100 with focal length 105 off creating at a focal plane 106 an image 102of an object 101, and it is not possible to discern three-dimensional(3D) information about the object 101 above or below the plane of focus.Any information above or below the plane of the object is not translatedto the plane of focus of the lens and is lost.

While other lenses can be added to the system to improve the imagequality or change the magnification, the 3D information is still lost.Holographic methods enable the imaging of the 3D information in a scene.A number of holographic techniques exist in which a sample isilluminated by a laser such that interference of light reflected oremitted from a sample in combination with a reference beam createsholograms which fully describe the 3D properties of an object [Nature161, 777-778 (1948)]. In classical holography a coherent source is splitinto a sample and reference beam, which then interfere with one anotherto create a hologram. These classical techniques, however, cannot beused to generate holograms from incoherent light. While these classicaltechniques cannot be used to measure incoherent light emissions, such asfrom a fluorescent sample, scanning holography has been proposed inwhich an interference pattern is scanned across a sample to excitefluorescence and then correlated with a sample beam to create a hologram[Opt. Lett. 22, 1506-1508 (1997)]. The scanning holography technique,however, is complex, and as a multibeam process it suffers fromstringent alignment requirements and is sensitive to environmentalinstability because of the need to prevent any vibration in the system.

Another technique for incoherent holography invented by one of thepresent inventors in 2006 [U.S. Pat. No. 8,542,421; Opt. Lett. 32,912-914 (2007)] is dubbed FINCH for Fresnel Incoherent CorrelationHolography. FINCH creates holograms from an object emitting incoherentlight in a single beam system by self-interference from two sphericalwaves originating from the object. Three example configurations of FINCHusing a spatial light modulator (SLM) are shown in FIG. 2 [adapted fromOpt. Exp. 19, 26249-26268 (2011)]. Described in FIG. 2 is 200 FINCH withtwo diffractive lenses displayed on the SLM 204, in which one (fa) ispositive and the other (f₂) is negative. The diffractive lenses focusthe light received from the object 101 through an intermediate lens 203into a hologram recorded by a CCD camera 206 at a distance 205 (z_(h))away from the SLM. Described in 201 is FINCH with two diffractive lenseson the SLM 204, in which both lenses are positive (f_(d) is the shorterfocal length, f₂ the longer). The remainder of this type of FINCH issimilar to that in 200. In 202 is a practical setup that emulates thesetup of 201, with one positive diffractive lens (f_(d)) displayed onthe SLM 204 and one positive glass lens 207 (f₂) placed near to the SLM.One skilled in the art will understand that in the previous paragraphand throughout this document, the SLMs or other elements that replacethe SLMs are not limited to displaying only one or two lenses, and thatthey may display three or more lenses or other phase patterns as desiredfor advantageous application to the holographic process.

FINCH has shown potential for fluorescence microscopy [J. Rosen and G.Brooker, “Non-scanning motionless fluorescence three-dimensionalholographic microscopy” Nat. Photonics 2, 190-195 (2008)], and much workhas been done to perfect the technique into a useful high resolution 3Dimaging technique. The concept that a 3D image could be obtained fromincoherent sources by a holographic process, without lasers, scanning oraxial translation or the need to capture images at multiple planes offocus to create a 3D image is appealing. The field has now advanced as aresult of additional work from the inventors [G. Brooker, N. Siegel, V.Wang, and J. Rosen, “Optimal resolution in Fresnel incoherentcorrelation holographic fluorescence microscopy,” Opt. Express 19,5047-5062 (2011); J. Rosen, N. Siegel, and G. Brooker, “Theoretical andexperimental demonstration of resolution beyond the Rayleigh limit byFINCH fluorescence microscopic imaging,” Opt. Express 19, 26249-26268(2011); B. Katz, J. Rosen, R. Kelner, and G. Brooker, “Enhancedresolution and throughput of Fresnel incoherent correlation holography(FINCH) using dual diffractive lenses on a spatial light modulator(SLM),” Opt. Express 20, 9109-9121 (2012); N. Siegel, J. Rosen, and G.Brooker, “Reconstruction of objects above and below the objective focalplane with dimensional fidelity by FINCH fluorescence microscopy,” Opt.Express 20, 19822-19835 (2012)] and others (P. Bouchal, J. Kapitan, R.Chmelik, and Z. Bouchal, “Point spread function and two-point resolutionin Fresnel incoherent correlation holography,” Opt. Express 19,15603-15620 (2011); X. Lai, Y. Zhao, X. Lv, Z. Zhou, and S. Zeng,“Fluorescence holography with improved signal-to-noise ratio by nearimage plane recording,” Opt. Lett. 37, 2445-2447 (2012); O. Bouchal andZ. Bouchal, “Wide-field common-path incoherent correlation microscopywith a perfect overlapping of interfering beams,” J. Europ. Opt. Soc.—Rap. Pub. 8, 13011 (2013)) including the demonstration that the FINCHoptical system is inherently super-resolving (J. Rosen, N. Siegel, andG. Brooker, “Theoretical and experimental demonstration of resolutionbeyond the Rayleigh limit by FINCH fluorescence microscopic imaging,”Opt. Express 19, 26249-26268 (2011); B. Katz, J. Rosen, R. Kelner, andG. Brooker, “Enhanced resolution and throughput of Fresnel incoherentcorrelation holography (FINCH) using dual diffractive lenses on aspatial light modulator (SLM),” Opt. Express 20, 9109-9121 (2012); N.Siegel, J. Rosen, and G. Brooker, “Reconstruction of objects above andbelow the objective focal plane with dimensional fidelity by FINCHfluorescence microscopy,” Opt. Express 20, 19822-19835 (2012)) Recentlyit has been shown that the reason for this is that FINCH overcomes theLagrange invariant (X. Lai, S. Zeng, X. Lv, J. Yuan, and L. Fu,“Violation of the Lagrange invariant in an optical imaging system,” Opt.Lett. 38, 1896-1898 (2013) [10]). More recently FINCH holograms havebeen created using electrically modulated transmission liquid crystaloptics (G. Brooker, N. Siegel, J. Rosen, N. Hashimoto, Makato Kuriharaand A. Tanabe, “In-line FINCH super resolution digital holographicfluorescence microscopy using a high efficiency transmission liquidcrystal GRIN lens,” Opt. Lett. 38(24), 5264-5267 (2013). Additionally,the inclusion of a Nipkow disk has been used to create confocal FINCHimages, (N. Siegel and G. Brooker, “Improved axial resolution of FINCHfluorescence microscopy when combined with spinning disk confocalmicroscopy,” Optics Express Vol. 22, pp 22298-22307 (2014) and U.S.patent application 62/023,958). The FINCH holographic process is thesubject of several patents including U.S. Pat. No. 8,009,340 issued onAug. 30, 2011; U.S. Pat. No. 8,179,578 issued on May 15, 2012; U.S. Pat.No. 8,405,890 issued on Mar. 26, 2013; U.S. Pat. No. 8,542,421 issued onSep. 24, 2014; and Japanese patentJP 5611588 issued on Sep. 12, 2014.

While FINCH is a considerable advance in incoherent holography, the SLMmethod of creating the two interfering beams still requires twodifferent lenses and those lenses require perfect alignment. Exampleembodiments of the invention disclosed in this application createoptically more perfect beams than any of the prior techniques forincoherent holography. Beams modulated by example embodiments do notsuffer from quantization error that is inherent in using quantizeddevices such as pixelated liquid crystal SLMs or Fresnel lenses or GRINlenses with discrete phase shifting regions and sharp boundaries betweenthe properties of neighboring regions. These errors include loss oflight into undesired diffraction orders, stepped instead of smooth phaseprofiles of the modulated beams, incomplete phase modulation,significant chromatic shift in focal lengths, and defects in the phaseprofiles of the modulated beams due to the mechanical structure of SLMs,GRIN lenses, etc. Beams modulated by some example embodiments may avoidall these defects, since these embodiments may not contain discreteregions with sharp boundaries (i.e. it is not quantized). There is nodiffraction off of mechanical frameworks and thus no loss to undesireddiffraction orders; and there is smooth continuous modulation of thephases of the modulated light; and there is only standard refractivechromatic dispersion error, which can be better corrected than thediffraction-induced chromatic dispersion. The SLM method used involvesdisplaying one or more different lens patterns on a spatial lightmodulator (SLM) [Opt. Lett. 32, 912 (2007); Opt. Exp. 19, 5047 (2011)]but may be prone to low hologram quality due to lens sampling and to lowefficiency due to higher-order diffracted images. These issues may leadto poor interference, high background and low resolution due to thelimited number of pixels and bit depth of the SLM. Furthermore, sinceSLM's are reflective, the optical arrangement requires that the SLM bepositioned on an angle from the optical axis of the imaging system orarranged on a beam splitter to circumvent mounting it on an angle.However, angled incidence of the original light beam makes calibrationof the SLM difficult for multiple focal lengths, and use of a beamsplitter significantly reduces the light budget of the optical system[Opt. Exp. 19, 5047 (2011)].

FIG. 3 shows a detailed schematic of a more recent method, which hasbeen to use a glass lens in conjunction with a liquid crystal Fresnellens or Gradient Refractive Index (GRIN or TLCGRIN) lens in a totallytransmissive arrangement, reported in Opt. Lett. 38, 5264-5267 (2013).On the left side of the FIG. 300 is depicted the detailed ray diagramfor a FINCH hologram of a point. The light leaves the object 101,traveling a distance 306 to be collected by the objective lens 301. Thecollimated light leaving 301 propagates the distance 307 to the first oftwo relay lenses, 302. The light travels the distance 308 to the secondrelay lens 303 and then a further distance 309 to the GRIN assembly 304.The GRIN assembly 304 with two effective focal lengths 312 and 313creates the two waves that propagate to the distances 310 and 311, whilethe hologram 305 is located at the plane removed from the GRIN assembly304 by the hologram distance 205. On the right side of the FIG. 301 isdepicted the detailed arrangement of the components in the referencedmicroscope system. All optics are centered on the optical propagationaxis 314. The dichroic beamsplitter and emission filter 315 and 316 arenecessary for fluorescence microscopy in order to introduce theexcitation light into the sample, and to separate the emission lightthat is received form any stray reflected excitation or other light,while the polarizing beamsplitting cube 317 is used to polarize thereceived light at an angle of 45 degrees to the active axis of the GRINassembly. The rejected polarization component from this polarizer issent to the camera 318 that records a standard image. The GRIN assembly304 contains a glass lens 319, and active GRIN 320 and an inactive GRIN321. The glass lens focuses all the light passing through it, while theactive GRIN adds additional focal length to the light that passesparallel to its axis, and the inactive GRIN serves to compensate forside effects of the light passing the active GRIN. Thus the two focallengths 312 and 313 are created. Distances are corrected to account forthe optical path through the glass of the BS cubes. The final two opticsare the phase shifting waveplate 322 and the output polarizer 323, whichmodulate the overall phase of the hologram and increase interferenceefficiency, respectively. The hologram plane 305 is between the twofocal lengths 312 and 313, and a camera 324 is used to record thehologram. FIG. 3 is adapted from Optics Letters 38, 5264-5267 (2013).

While the TLCGRIN method is an improvement over the SLM, it still islimited by the reduced imaging quality of a Fresnel lens or the limitednumber of graded regions used to create a liquid crystal GRIN lens.Furthermore it is challenging to make GRIN lenses with sufficientaperture and shortness of focal length for high quality imaging andcompactness of a holographic system. In this GRIN lens system example,the GRIN lens had a 5000 mm focal length and the glass lens a 300 mmfocal length. Furthermore both the SLM and GRIN lens systems requireelectrical control of the devices in addition to compensating lenses tocontrol for dispersion in the liquid crystal material. The TLCGRINmethod requires external power to induce the birefringent effect ofdifferential modulation of different polarization components of thereceived light. Since the GRIN lens has multiple rings concentricallyarranged around its center, each of which has a discrete constant phaseshift value with relatively sharp boundaries between rings, it isquantized, though it is not as severely quantized as an SLM. Thiscombination of focal lengths creates a spacing factor between the twofocal lengths of less than 3%, which reduces the axial depth of 3Dobjects that can be reliably imaged by the holographic system [Opt. Exp.20, 9109 (2012)].

To address this, the inventors have discovered a unique use forspherical lenses that can be constructed of birefringent materials. FIG.4 shows an example of a lens 400 made from a birefringent material,according to some embodiments. Birefringent substances have two distinctpolarization sensitive refractive indices and thus lenses made from suchmaterials always have two focal lengths f₁ 401 and f₂ 402 and produceblurry images when randomly polarized light is passed through them,since a single sharp plane of focus is not possible unless the image isviewed through a polarizer. When randomly polarized light is passedthrough the lenses, a single sharp focus cannot be obtained since themultiple refractive indices of the material cause the lens to display adifferent focus for light of p or s polarization, creating two images atdistances 403 and 404. Thus these lenses yield a doubled or blurry image405, which is generally undesirable in standard optical applications.For this reason birefringent materials are not typically used to makeoptical lenses because of this ordinarily undesirable property; evidenceof this is that birefringent lenses are not readily commerciallyavailable from optical supply houses. Currently birefringent lenses mustbe custom made and there are few reports in the literature of theirconstruction [Proc. of SPIE Vol. 6018, 601812 (2005); Meas. Sci.Technol., 17, 1367 (2006); Optik 118, 335-339 (2007)]. However sincebirefringent materials such as calcite, barium borate, lithium niobateand quartz can be readily worked just like glass, it is possible toreadily prepare lenses of birefringent materials to any lensspecification, given a rationale for making them.

The inventors have discovered that the simultaneous usage of themultiple focal lengths of birefringent lenses can be very advantageousto create very high quality holograms that can reveal the threedimensional information of objects. Embodiments of the invention can beapplied to many forms of holography including FINCH and operates in anelectrically independent manner with optical characteristics that yieldunmatched holographic image quality which exceeds the performance ofstandard imaging methods. Furthermore, in addition to holographicimaging applications, the embodiments also enhance and simplify otherforms and uses of holography and interferometry. For an example,birefringent lenses were already found in nature long ago in the eye ofthe trilobrite, a creature that lived in the sea 450 million years ago.These eye lenses were called schizochroal and made of birefringentcalcite. One might speculate that lenses made of calcite became extinctduring evolution because of their undesirable optical properties.Calcite is an optically clear material with two different refractiveindices depending upon the plane of polarization. Even though it is nota good material to make standard lenses, its polarizing properties areexploited to make polarizers and polarization sensitive devices such asGlan-Taylor prisms. Calcite is used because it is optically clear andits crystal structure can efficiently pass a single axis of linearpolarization. However if lenses are made of calcite, because of thedifferent refractive indices at the two planes of polarization, twodistinct polarization sensitive focal lengths of those lenses areobserved (seehttps://community.dur.ac.uk/g.d.love/downloadable/china05.pdf). Howeverwith mixed polarization light, which is the common form of light in theenvironment, a blurred image would result if lenses were made ofbirefringent materials. While the trilobrite had calcite for its lensmaterial, one might wonder if its vision was blurred or if it could seethe two focal planes because its photoreceptors were cross polarized.

However, an imaging method that required different aligned copies of thesame image could benefit greatly from just such a birefringent lens.Incoherent holography, a class of holography that includes FINCH andother methods [Opt. Lett. 32, 912 (2007); Nat. Photonics 2, 190 (2008);Opt. Express 19, 5047 (2011); Opt. Express 19, 26249 (2011); Opt.Express 20, 19822 (2012); Opt. Lett. 38, 3922 (2013); Opt. Lett. 38,5264-5267 (2013), and U.S. Pat. Nos. 8,009,340, 8,179,578 and8,542,421], is a technique for creating holograms from the interferenceof two copies of the same image, or from any single EM radiation wavethat is split into two copies, and has been demonstrated usingpolarization-sensitive optical elements (PSOEs) such as SLMs and liquidcrystal Fresnel and GRIN lenses. These PSOEs, which are not classicalrefractive spherical lenses but which may be diffractive or refractivein operation, serve to split the image beam into two parts withdiffering spherical curvatures. In the further description of theprocess in relation to embodiments, the inventors consider lightemanating (by emission or reflection or any other process) from a singleinfinitesimally small object point, which creates a “point hologram”that suffices to describe the system; extended objects larger than thiscreate holograms that are simply the sums of the holograms of all thediffering points constituting the extended object. A broad, collimatedlaser beam may be used as a model source of EM radiation in thesesystems, since the image of such a beam is a diffraction-limited spot asfrom an infinitesimal point source. This aspect enables the empiricalcharacterization of the best response of any such system.

FIG. 5 shows a schematic of the FINCH process highlighting the role ofthe PSOE. The PSOE 501 has two different focal lengths, of which f_(d1)is the shorter and f_(d2) is the longer. Other optical elements orgroups 500, 502 may be used to make specific alterations in the overallphase, polarization, aberration correction or magnification or hologramsize of the system, but the beam separation is solely a result of theuse of the PSOE. After emanating from the object and possibly passingother optical elements, the light wave is split into two waves, ofdiffering focal lengths by the PSOE. This splitting can be accomplishedby reflecting off of or transmission (e.g., by refraction ordiffraction) through the PSOE. These waves propagate through the samespace in the same direction, and are termed the signal and reference orf_(d1) and f_(d2) waves. Currently this is accomplished in one of twoways:

-   -   1. By polarization: the received wave hitting the PSOE is        polarized at 45 degrees to the polarization axis of the PSOE.        Thus half of the wave with polarization component projected        parallel to the PSOE polarization axis is given the curvature        encoded in the PSOE, while the half of the wave with        polarization component projected perpendicular to the PSOE        polarization axis maintains its original curvature. The result        is the f_(d1) and f_(d2) waves.    -   2. By sampling of the PSOE: The PSOE is divided into more than        one portion, each of which is encoded with differing spherical        phases. The portions may be interspersed with each other and not        contiguous. The received wave hitting the PSOE is polarized        entirely parallel to the PSOE polarization axis, and the wave        emerging from the PSOE has different portions with differing        curvatures added corresponding to the curvatures encoded in the        different portions of the PSOE. If the PSOE has two portions,        the two wave portions emerging from the PSOE are termed f_(d1)        and f_(d2). However the PSOE can have more than two portions, in        which case there are light waves termed f_(d3), etc.

Current technologies serving as polarization-sensitive PSOEs to generatethe f_(d1) and f_(d2) waves include digital spatial light modulators(SLMs), liquid crystal (LC) Fresnel lenses and LC gradient refractiveindex (GRIN) lenses. In some configurations these components are alsoused in conjunction with classical lenses, or more than one of thecomponents may be used in conjunction with each other.

After propagating from the PSOEs, the two waves interfere and create thehologram recorded at the detector (z_(h)) plane. The detector may be aCCD, CMOS or other camera or image capture device as well as a pointdetector or solid-state device such as an avalanche photodiode.Optionally the waves may pass through a variable phase shifter and apolarizer. To reconstruct a point or image and provide the basis toremove bias and the twin image in holography, the detector captures twoor more raw holograms, in which the phase of one of the beams is set todiffer by a predetermined amount in subsequent raw holograms, to allowfor the recovery of the complex hologram that fully captures the phasecharacteristics of the original EM source [Optics Letters 22(16),1269-1270 (1997)]. The collection of raw holograms with such differentphase factors is critical to achieving the optimal result with FINCH andsimilar holography methods.

One of the key parameters in this process is the relationship betweenthe focal lengths f_(d1) and f_(d2) and the hologram recording plane atz_(h). Holograms may be recorded at any point after the PSOE, but theoptimal hologram quality is made possible when the two waves obey acondition of maximal spatial overlap. The condition to ensure maximumoverlap between the f_(d1) and f_(d2) beams is met when the hologram isrecorded at the plane

$\begin{matrix}{z_{h} = {\frac{f_{d\; 1}f_{d\; 2}}{( {f_{d\; 1} + f_{d\; 2}} )}.}} & (1)\end{matrix}$

This relationship may also be expressed as

z _(h)=(1+s)′f _(d1)=(1−s)′f _(d2),  (2)

where the spacing factors obeys the equality:

$\begin{matrix}{s = {{\frac{f_{d\; 2} - f_{d\; 1}}{f_{d\; 2} + f_{d\; 1}}}.}} & (3)\end{matrix}$

As s increases (the distance between f_(d1) and f_(d2) increases), thepoint hologram at the optimal z_(h) plane also increases in size, asdescribed by the following equation:

R _(H) =s′R ₀,  (4)

where R_(H) is the aperture radius of the hologram and R₀ is theaperture radius of the wave at the PSOE or equivalent. This sizeincrease renders the point hologram more easily resolvable by recordingdevices but decreases the peak intensity of the hologram. There areother factors [Opt. Express 20, 9109 (2012)] that also establish upperand lower bounds for s. It is very desirable to have complete controlover s over a wide range in order to be able to optimize the holographicsystem for all possible variables such as magnification of the image,spatial size of the point hologram, fringe spacing and number of fringestherein, and intensity of the light at the hologram plane. The s factormay not itself change the resolution of the image coded by the hologram,but does affect the ease with which the hologram may be recorded; andfurther, any arrangement used to change s may affect other image factorssuch as magnification and depth of field. In some aspects, thecapability provided in certain example embodiments to vary the s factor,yields the benefit of the configurability available in the SLM-basedholography techniques while yielding higher quality interferencepatterns than any GRIN-based holography techniques.

Each of the three current technologies mentioned above can serve tocreate f_(d1) and f_(d2) by reflection off of or transmission throughthe PSOE, but each also bears significant disadvantages:

-   -   1. SLMs are easily adjustable to produce different focal length        PSOEs at will, in the form of digitized Fresnel phase patterns,        but suffer from low focusing efficiency to the desired image, as        diffraction from the pixilated digital SLM causes significant        light loss into transverse foci of higher diffraction orders.        Additionally, the PSOEs created on SLMs suffer from significant        variability in focal length as a function of light wavelength        (an effect termed chromatic aberration) which may degrade        performance in hologram formation.    -   2. LC Fresnel lenses are polarization sensitive and do not        suffer from higher-order transverse foci, but may display other        axial foci and certainly suffer from significant chromatic        aberration. They are also not adjustable, and offer only a        single nominal focal length.    -   3. LC GRIN lenses have focal lengths adjustable as a function of        applied voltage, and less chromatic aberration than SLMs or LC        Fresnel lenses, but have very long focal lengths that require        them to be paired with regular refractive lenses in order to        achieve reasonable overall focal lengths. Even when combined        with refractive lenses, LC GRIN lenses offer limited        possibilities for spacing factor. Finally, currently used LC        GRIN lenses are quantized approximations of lenses (because of        the practical limitation of the number of differentially        refractive zones possible) and thus impose spatial distributions        of light in the unfocused beams that can cause reduced        interference efficiency and accuracy of focal length        calculation.

There is a pressing need in this field for the introduction of a deviceto create the f_(d1) and f_(d2) beams with equivalent quality to that ofa spherical refractive lens and without the disadvantages mentionedabove, and with increased flexibility in the spacing factor s.Birefringent materials possess two or more refractive indices alongdifferent propagation directions in the material, termed the ordinaryand extraordinary axes. These axes have refractive indices denoted n_(o)and n_(e), respectively. Since the focal length of a lens is dependentin part on the refractive index of the material comprising the lens,these materials can be used to create spherical lenses that possess twodifferent polarization-dependent focal lengths, each of which produces aspherical beam and a focal spot of equal quality to those of a standardglass lens. FIG. 6 shows a schematic of a birefringent lens (BRL)focusing light of differing polarization to different focal planes. FIG.6a 600 shows a cross-section of a BRL, with the ordinary 602 andextraordinary 603 refractive indices projected along the x and yCartesian axes of the lens. FIG. 6b 601 shows the focal lengths f_(be)606 and f_(bo) 607 of the single birefringent lens (with radii ofcurvature R₁ 604 and R₂ 605 for the two surfaces of the lens) for lightpolarized parallel to the extraordinary axis and for light polarizedparallel to the ordinary axis of the lens, respectively. FIG. 6b shows aconvergent lens 400. FIG. 6c shows the focal lengths f_(be) 608 andf_(bo) 609 of the single birefringent divergent lens 613 (with radii ofcurvature R₁ 610 and R₂ 612 for the two surfaces of the lens). Thequality of the beams and the focal spots of the BRL is much improvedover those from diffractive PSOEs mentioned above. A perfect FINCH pointhologram is composed of many well-modulated spherical fringes followingthe sinusoidal Fresnel zone plate, in which the fringes are allperfectly spherical, concentric with fringe size that decreases inproportion to distance from the center, and in which the dark fringes donot contain any light at all; the maximum quality FINCH hologram of areal object is obtained as the sum of many point holograms originatingfrom the different points of the object. To obtain a perfect or nearlyperfect FINCH hologram it may be necessary that a reference and samplebeam path interfere such that the image size is identical or nearlyidentical for both beams at a hologram plane. This can be readilyaccomplished by adjusting the focal lengths and shape of thebirefringent lenses. In FIG. 6 a schematic is shown where the beamsintersect at a plane between the focal lengths of the birefringent lens.Birefringent Refractive Lenses used in example embodiments, such asthose shown in FIG. 6, offer advantages over PSOEs in several aspects ofincoherent hologram generation, including:

-   -   1. Elimination of the noise and image artifacts due to unwanted        diffraction orders of PSOEs or the quantization error inherent        in digital or binary representations of lenses.    -   2. The possibility of correction of chromatic, spherical and        other aberrations by use of corrective optics including        non-birefringent and birefringent optics.    -   3. Precise and flexible tailoring of the spacing factors by        choice of BRL material, curvature and associated optics.    -   4. Simplification of and size reduction of the optical assembly        by removal of electronic and reflective components.

Some example embodiments of the invention covers, at least in part, theuse of a BRL, alone or in conjunction with other refractive lenses orother optical elements, to effect the splitting of the received waveinto two orthogonally polarized waves with differing spherical curvatureto create holograms. Birefringent crystals have differing refractiveindices along their ordinary and extraordinary crystal axes, and bycutting (and/or grinding and polishing) a lens from such a material inthe proper orientation with these two axes perpendicular to each otherand both lying in the plane of the lens orthogonal to the direction oflight propagation through the lens, a refractive lens with specialproperties may be created. These special properties are that the lensfocuses light polarized parallel to one of its polarization axes (forexample, the ordinary axis, also identified here as the x axis in aCartesian system) to a given focal plane, while the light polarizedparallel to the other axis (the extraordinary or y-axis) is focused to adifferent focal plane (see FIG. 6). This may be easily understood byreferring to the thin lens equation:

$\begin{matrix}{{\frac{1}{f} = {( {n - 1} )( {\frac{1}{R_{1}} - \frac{1}{R_{2}}} )}},\mspace{14mu} {{{or}\mspace{14mu} f} = {{\frac{1}{( {n - 1} )}( \frac{R_{1}R_{2}}{R_{2} - R_{1}} )} = \frac{R_{eff}}{( {n - 1} )}}},} & ( {5a} ) \\{{\frac{1}{f} = {( {n - 1} )( \frac{1}{R} )}},\mspace{14mu} {{{or}\mspace{14mu} f} = {\frac{R}{( {n - 1} )} = \frac{R_{eff}}{( {n - 1} )}}},} & ( {5b} ) \\{R_{eff} = {\begin{matrix}{\frac{R_{1}R_{2}}{R_{2} - R_{1}},} & {{for}\mspace{14mu} a\mspace{14mu} {lens}\mspace{14mu} {with}\mspace{14mu} {two}\mspace{14mu} {curved}\mspace{14mu} {sides}} \\{R,} & {{for}\mspace{14mu} a\mspace{14mu} {lens}\mspace{14mu} {with}\mspace{14mu} {one}\mspace{14mu} {curved}\mspace{14mu} {side}}\end{matrix}}} & ( {5c} )\end{matrix}$

with f being the focal length of the lens, n the refractive index of thelens material, R₁ and R₂ the radii of curvature of the two sides of thelens, and R_(eff) is the “effective” total curvature of the lens.Equation 5b is for the specific case of a lens with one flat side(plano-concave or plano-convex) and one curved side with curvature R. Ascalled out in equation 5c, R_(eff) for a lens with two curved sides isexactly equivalent to R of a plano-concave or plano convex lens.Equivalently to using a solid birefringent crystal, a birefringentliquid crystal material may be used to create a BRL when aligned andplaced between two substrates with curvatures R₁ and R₂. Thus a singleBRL, made from birefringent material with n_(o) and n_(e) for theordinary and extraordinary refractive indices, has focal length f_(bo)for light polarized along its ordinary axis and focal length f_(be) forlight polarized along its extraordinary axis. By virtue of theextraordinary axis of the lens being orthogonal to the direction oflight propagation, the extraordinary axis will not impart a transverseoffset to the beam as can happen in other axis orientations. The twofocal lengths of the BRL may be used as the two focal lengths necessaryfor the holographic process, i.e. f_(be) and f_(bo) may be substitutedfor f_(d1) and f_(d2) in equation 3. By reference to equation 3, then,any single lens made of a given type of birefringent material will havea constant spacing factor no matter the physical curvatures of the lens:

$\begin{matrix}{s = {{\frac{f_{be} - f_{bo}}{f_{be} + f_{bo}}} = {{\frac{n_{o} - n_{e}}{n_{o} + n_{e} - 2}}.}}} & ( {6a} )\end{matrix}$

Equation (1) may be simplified as follows for a birefringent lens:

$\begin{matrix}{z_{h} = {{2\frac{f_{be}f_{bo}}{f_{be} + f_{bo}}} = {\frac{2R_{1}R_{2}}{( {R_{2} - R_{1}} )( {n_{o} + n_{e} - 2} )}.}}} & ( {6b} )\end{matrix}$

However, when used in conjunction with a non-birefringent lens, each ofthe focal lengths of the birefringent lens combines with the singlefocal length f_(r) of the non-birefringent lens to result in two newcombined focal lengths, one for each polarization axis of thebirefringent lens. Under the thin-lens approximation and assuming n_(o)distance between the birefringent lens and the standard lens, the focallengths

and

of the combined system are now:

$\begin{matrix}{{= \frac{f_{be}^{\prime}f_{r}}{f_{be} + f_{r}}},{{{and}\mspace{14mu} } = \frac{f_{bo}^{\prime}f_{r}}{f_{bo} + f_{r}}},} & (7)\end{matrix}$

and the combined spacing factor s¢ of the hologram system can beincreased and decreased from this constant value according to thefollowing equation:

$\begin{matrix}{{= {{\frac{-}{+}} = {\frac{f_{be} - f_{bo}}{f_{be} + f_{bo} + \frac{2f_{be}f_{bo}}{f_{r}}}}}},} & ( {8a} )\end{matrix}$

and correspondingly from Equation (1)

$\begin{matrix}{= {{2\frac{}{+}} = {\frac{2R_{1}R_{2}}{{( {R_{2} - R_{1}} )( {n_{o} + n_{e} - 2} )} + \frac{2R_{1}R_{2}}{f_{r}}}.}}} & ( {8b} )\end{matrix}$

Note the similarity of the right-most part of equation 8a to theinternal part of equation 6a, showing the additional factor foradjustment of the spacing factor. Table 1 contains the refractiveindices, curvatures, focal lengths and inherent spacing factors ofspherical lenses that could be made from several select birefringentmaterial, calculated from equations 4-6, as well as correspondingaltered focal lengths and altered spacing factors for systemsincorporating these lenses and select glass lenses, calculated fromequations 7 and 8. The collected data demonstrate the possibility toexercise total control of the spacing factor and other holographyproperties of BRL based systems. Some example embodiments allow thespacing factor to be freely altered between 0.001-0.33, for example,while maintaining perfect beam overlap, for the purposes of adjustingthe intensity of and number of fringes in the point hologram.

TABLE 1 Refractive indices, curvatures, focal lengths and incoherenthologram parameters of selected birefringent materials. Birefring. R₁ R₂f_(bo) f_(be) z_(h) f_(r) f_(bo)′ f_(be)′ z_(h)′ material n_(o) n_(e)(mm) (mm) (mm) (mm) S (mm) (mm) (mm) (mm) s′ (mm) Calcite 1.66 1.49 95−95 72 98 0.150 83 −166  128 237 0.300 166 Calcite 1.66 1.49 190 −190144 195 0.150 166 N/A 144 195 0.150 166 Calcite 1.66 1.49 380 −380 289391 0.150 332 332 154 179 0.075 166 Quartz 1.54 1.55 95 −95 87 86 0.00887 −173  176 170 0.016 173 Quartz 1.54 1.55 190 −190 175 172 0.008 173N/A 175 172 0.008 173 Quartz 1.54 1.55 380 −380 349 344 0.008 346 346174 172 0.004 173 barium 1.68 1.55 95 −95 70 86 0.101 77 −200  108 1500.164 126 borate barium 1.68 1.55 190 −190 140 172 0.101 154 N/A 140 1720.101 154 borate barium 1.68 1.55 380 −380 280 343 0.101 309 100 74 770.025 76 borate The first column refers to the birefringent material ofthe lens discussed in the row. n_(o) and n_(e) are the ordinary andextraordinary refractive indices of the birefringent material. R₁ and R₂are the radii of curvature of the birefringent lens. f_(bo) and f_(be)are the ordinary and extraordinary focal lengths of the birefringentlens, as discussed in the text. s is the inherent spacing factor of thebirefringent material, as discussed in the text. z_(h) is the optimalhologram distance for the given combination of birefringent material andlens curvature, as discussed in the text. f_(r) is the focal length ofan optional non-birefringent lens used in conjunction with thebirefringent lens for the purpose of altering the spacing factor andoptimal hologram distance. f_(bo)′ and f_(be)′ are the altered ordinaryand extraordinary focal lengths of the birefringent lens, as discussedin the text. s′ is the altered inherent spacing factor of thebirefringent material, as discussed in the text. z_(h)′ is the alteredoptimal hologram distance for the given combination of birefringentmaterial and glass lens, as discussed in the text.

The implications of equation 8 include that:

-   -   1. The choices of R₁ and R₂ of the birefringent lens and focal        length f_(r) of the standard lens allow any spacing factor to be        achieved with a BRL made from any birefringent material. This is        illustrated in Table 1, showing that for any given birefringent        material, the spacing factor s_(s) is an intrinsic property, but        that the spacing factor s¢ of the combination of a birefringent        lens and a non-birefringent lens may be adjusted up or down. The        focal lengths f_(r) of the non-birefringent lens in Table 1 were        chosen to result in sets of lens combinations with the same        z_(h) but different s¢ (for the calcite and quartz birefringent        lenses), or to show changes in both z_(h) and s¢ (barium borate        birefringent lens).    -   2. Use of a positive lens as the standard lens will reduce s′ as        compared to s, while use of a negative lens as the standard lens        will increase s′ as compared to s.    -   3. Hybrid lenses of any desired focal length, achromaticity and        spacing factor can be made of materials that are composed of        birefringent and non-birefringent material components cemented        together.    -   4. While compound lens compositions of birefringent materials        can make a device achromatic, it should be realized that the        wavelength specific refraction of each lens in a non-achromatic        birefringent lens will proportionally shift the focus of each of        the lens focal points made from a birefringent material. Thus        the plane of maximum interference will be shifted depending on        wavelength. Because of this, a feature enabled by using        birefringent lenses is that wavelength specific holograms can be        obtained by hologram detection at any of those wavelength        specific hologram planes even though the input is polychromatic.        FIG. 7 shows an example of the shift in hologram planes 700,        701, 702 as a function of variance in wavelength. Dashed lines        and double lines represent a blue wavelength 700, dashed single        dot lines and solid lines represent a green wavelength 701, and        dashed double dot lines and triple lines represent a red        wavelength 702.

One skilled in the art will understand that the above equations 5, 7 and8 may be adjusted for use with more accurate lens equations and toaccount for some distance between the BRL and the glass lens.

Thus birefringent refractive lenses can be used to significantlymaterially improve hologram creation when used in the followingconfigurations:

-   -   1. As the sole lens or optical element involved in hologram        formation.    -   2. In conjunction with another paired lens or optical element to        alter the spacing factor of the f_(d1) and f_(d2) beams, where        the other lens or optical element may consist of:        -   a. A single lens or optical element.        -   b. A compound lens or optical element.        -   c. A sequence of lenses or optical elements.    -   3. In conjunction with another corrective lens or optical        element designed to correct spherical, chromatic or other        aberrations in the birefringent refractive lens, where the        corrective lens or optical element may consist of:        -   a. Single, compound or multiple standard non-birefringent            corrective lenses or optical elements designed to correct            the aberrations of one or the other focal lengths of the            birefringent refractive lens.        -   b. Single, compound or multiple standard non-birefringent            corrective lenses or optical elements designed to correct            the average aberration of the two focal lengths of the            birefringent refractive lens.        -   c. Single or multiple birefringent corrective lens or            optical element designed to correct the aberrations of one            or the other focal lengths of the birefringent refractive            lens, in which the corrective birefringent lens may be made            of a different birefringent material than the            hologram-forming birefringent refractive lens.        -   d. Single or multiple birefringent corrective lens or            optical element designed to correct the average aberration            of the two focal lengths of the birefringent refractive            lens, in which the corrective birefringent lens may be made            of a different birefringent material than the            hologram-forming birefringent refractive lens.        -   e. Single or multiple birefringent corrective lens or            optical element designed to correct the aberrations of one            or the other focal lengths of the birefringent refractive            lens, used in conjunction with standard non-birefringent            lenses or optical elements, in which the corrective            birefringent lens may be made of a different birefringent            material than the hologram-forming birefringent refractive            lens.        -   f. Single or multiple birefringent corrective lens or            optical element designed to correct the average aberration            of the two focal lengths of the birefringent refractive            lens, used in conjunction with standard non-birefringent            lenses or optical elements, in which the corrective            birefringent lens may be made of a different birefringent            material than the hologram-forming birefringent refractive            lens.    -   4. In conjunction with both paired and corrective lenses or        optical elements of any of the kinds listed in items 2 or 3 of        this list

Experimental work has confirmed the improvement seen in a FINCH systemwhen a current TLCGRIN-based system was compared with a BRL-basedsystem. FIG. 8 shows FINCH holograms obtained using a laser as the EMradiation source, from a FINCH system configured with liquid crystalGRIN lenses and a glass lens to create two focused beams with a hologramplane between them, as in the prior art shown in FIG. 3. The top threepanels 800, 801, 802 in FIG. 8 show three phase-shifted raw FINCHholograms, which are significantly distorted from the well-modulatedspherical Fresnel patterns that should characterize the ideal responseof a FINCH system. The bottom three panels in FIG. 8 show, from left toright, the magnitude 803 of the complex FINCH hologram, the phase 804 ofthe complex FINCH hologram, and finally the reconstructed image 805 ofthe laser beam. The magnitude shows large intensity fluctuations andboth the magnitude and phase show deviations from a perfect sphericalshape. The reconstructed spot shows significant background signal aswell as deviations from a perfect point shape. FIG. 9 shows the resultsfrom a similar system in which the major difference was the use of aspherical calcite BRL to induce the differing phase properties betweenthe signal and reference beams instead of a GRIN lens plus glass lensarrangement; an imaging relay lens was also used to project the hologramonto the camera after it passed the BRL. All other factors and settings,including light source, ancillary optics, polarizers, phase shiftingplate and voltage, and cameras were the same as those used to produceFIG. 8. In the top row of FIG. 9a are shown three phase shifted rawholograms 900, 901, 902 as in the top three panels of FIG. 8. The rawholograms are nearly perfect representations of the desired sphericalFresnel pattern, and show many more Fresnel rings than the raw hologramsin FIG. 8, a result of the much greater spacing factors possible whenusing a calcite BRL instead of the GRIN/glass system. In the bottomthree panels of FIG. 9a , we again see, from left to right, the complexhologram magnitude 903 and phase 904 and the reconstructed image 905 ofthe laser. The magnitude and phase are both perfectly sphericalpatterns, with the magnitude free from the significant intensityfluctuations that affect the system described in FIG. 3 and used toproduce FIG. 8. The phase shows a smooth slope and neat transitions atphase wrapping regions, and the reconstructed spot is point-like andfree from excessive background levels. The dramatic improvement of FIG.9a over FIG. 8 is indicative of the overall improvement in holographicimaging that BRLs can provide over other PSOEs. It should be noted thatthe differences in hologram diameter are caused by the fact that the sof GRIN-based FINCH is about 0.03, while that of calcite FINCH isapproximately 0.11, and that this along with the secondary relay ofcalcite FINCH affects the size of the reconstructed spot, rendering thatquantity alone insufficient for judging the performance of thebirefringent lens relative to the GRIN lens. However the improvements inthe raw hologram symmetry and complex hologram intensity and phase arenot dependent on s and do clearly indicate the advantages of thebirefringent lens in producing FINCH interference.

Birefringent spherical lenses made from alpha-barium borate (α-BBO oralpha-BBO) were also used in some embodiments to create FINCH images ofstandard objects in fluorescence microscopy. Birefringent lenses andoptical flats of calcite and of α-BBO were made, according to anembodiment, by standard methods for fabrication of optical glasscomponents, with their extraordinary axes lying in the plane orthogonalto the direction of light propagation through the optic. Birefringentoptics may be made from α-BBO because of its temperature andenvironmental stability as well as the property that it can be grown inlarge single crystals with high optical quality. In a microscopeconfigured in a manner similar to FIG. 3, with a BBO lens and a separateBBO compensating flat replacing the active and inactive GRIN lenses 320and 321, a fluorescent USAF test pattern and a sample of 100 nm diameterbeads were imaged by both classical and FINCH imaging, the results ofwhich are shown in FIG. 9b and FIG. 9c . FIG. 9b shows the results ofclassical imaging 912 and FINCH with BBO (e.g., α-BBO) lens imaging 914.A 20×0.75 NA Nikon and a 60×1.49 NA Nikon TIRF objective lens were used,respectively, for the USAF pattern and the 100 nm beads. The wide-fieldand FINCH images of the 100 nm beads (590 nm wavelength) weredeconvolved using a commercial application developed by Microvolution,Inc. Blind deconvolution was applied, using as the initial PSF guesses aclassical PSF for the wide-field image and a custom PSF for the FINCHimages. As shown in the images in FIG. 9b and FIG. 9c , the BBO-basedFINCH imaging microscope was able to image an extended object atresolution comparable to that reported in the literature for aGRIN-based FINCH system. Furthermore, analysis of the images of 20randomly selected bead images (shown in FIG. 9c for widefield 916 andBBO-based FINCH 918) from across the imaging plane show that theBBO-based FINCH system was able to resolve the beads at better thanclassical resolution limit, as predicted for FINCH imaging. In data notshown, a 100×1.4 NA Nikon objective was used to image a sample ofFITC-stained microtubules, and the FINCH images of the microtubulesshowed cross sections of approximately 120 nm, further demonstrating theefficacy of the embodiment. FIG. 9d results, however, were obtained fromα-BBO-FINCH imaging of 110 nm sub-resolution fluorescent beads whichrequire higher resolving power. 920 and 922 are 8×8 μm zoomed selectionsof 110 nm fluorescent beads for resolution comparison of the same areabetween 920 widefield fluorescence and 922 α-BBO-FINCH images. 924 and926 are 1 μm square zoomed images of the same randomly selected beadsfrom 920 and 922 respectively. The beads in the respective parts of 924and 926 are the same. A histogram 928 of FWHM size distributions amongstthe 20 beads that were measured in 924 and 926, shows an approximatelytwo-fold reduction in FWHM by FINCH. The plots 930 depict the averageFWHM sizes of the 110 nm beads as measured by fluorescence andα-BBO-FINCH microscopy, with normalized Gaussian functions of theaverage width measured from the 20 selected beads. When these beads wereimaged with FINCH using a Nikon 60×1.49 NA TIRF objective (not in TIRFgeometry), the α-BBO-FINCH measurements of the bead sizes averaged149±11 nm, significantly smaller than the 287±20 nm average from thecorresponding classical images of the same exact beads. These resultsare comparable to beads imaged at a shorter wavelength by anothersuper-resolution technique. To the inventors' knowledge these are thesmallest objects that have been measured by any self-referencedholographic method, as well as the first demonstration of super-resolvedself-referenced holographic imaging of any kind with a highmagnification, high NA system. These advances are due to the highimaging quality of the birefringent lens incoherent interferometer basedFINCH system of example embodiments. The development of the singlecrystal birefringent lens for FINCH holographic imaging enables FINCH toreach its full potential at the highest resolution and magnification and

achieve the theoretically predicted super resolution not possible withother previously used hologram forming approaches. This is because ofthe common-path simplicity of the FINCH method and flexible,nonquantized polarization-based beamsplitting quality of thebirefringent crystal lens approach that is not achievable with SLMs,currently available liquid crystal lenses or even dual beam-pathinterferometers that have also been used to generate self-referencedholograms. This achievement shows the potential of birefringent crystallenses for use in other holographic and interferometric methods as well.For example these lens interferometers could simplify and stabilize thelaser generated excitation beam in structured illumination or scanningholography as well as other incoherent interferometric applications.

Other systems may be constructed that make use of BRLs. As shown in FIG.10, another system 1000 incorporates two BRLs 400 and 1002 usedtogether, whether said BRLs are made from the same material or not, toachieve further modification of the two waves. The cross sectiondiagrams 1001 of the two BRLs show how a second BRL 1002 could be used,with its axes 702 and 703 parallel or perpendicular to the correspondingaxes of the first BRL 400, to provide chromatic, spherical or othercorrections to the first BRL.

FIG. 11 shows another system 1100 incorporating a BRL with two flatsides, hereinafter called a birefringent flat (BRF) 1102, acting as aphase-delay compensating optic to change the total optical pathdifference between the two waves in addition to the BRL thatdifferentially changes the spherical curvature of the wavefronts of thewaves. The cross sections 1101 show the relative orientations of theordinary and extraordinary refractive indices 702 and 703 of the the BRL400 and the BRF 1102. Optical path length (OPL) is a measure of thedistance traveled by an EM wave, taking into account both thethicknesses of various media the waves traverse as well as theirrefractive indices:

OPL=Σd _(i) n _(i)  (9)

where d_(i) and n_(i) are the thicknesses and refractive indices of allmedia in the path traveled by the wave. The optical path difference(OPD) of two waves is a measure of the difference in the OPLs the wavestraveled. When dealing with incoherent holography, it is important tokeep the total optical path difference between the two waves low inorder to maintain the conditions necessary for holography interferenceto occur. The difference is required to be less than the coherencelength of the light, which is generally approximated as λ²/Δλ, where λis the center wavelength and Δλ is the bandwidth. In the microscopyrealm, the coherence length is on the order of 10 μm, at least an orderof magnitude shorter than the lasers or monochromatic light to whichprevious interferometers with birefringent lenses have been restricted.The BRL not only imparts different curvatures to the two waves throughthe two focal lengths f_(be) 606 and f_(bo) 607, but also imparts anoverall optical path difference OPD_(o) between the two waves that isproportional to the thickness d_(BRL) of the BRL and the two refractiveindices of the birefringent material:

OPD _(o) =d _(BRL)(n _(o)-n _(e))  (10)

In any form of FINCH, the OPD between the two differentially focusedbeams has a geometric component due to the different physical paths thatthe light waves travel after exiting the differential focusing opticthat is less than the coherence length and thus does not prevent thewaves from interfering. For the GRIN method the birefringence|Δn|=|n_(o)−n_(e)| of the liquid crystal material in the GRIN lens isenough to cause an additional large OPD component that is greater thanthe coherence length, which must be compensated for by another optic ifinterference is to be observed. A similar effect occurs in this case, inwhich the birefringent lens not only imparts different phase curvaturesto the two waves through the two focal lengths f_(be) and f_(bo),related to the curved surfaces of the lens, but also imparts an overalloptical path difference ΔOPD between the two waves that is proportionalto the thickness d_(BRL) of the central cross-sectional part of thebirefringent lens as in equation (10). This ΔOPD does not contribute tothe desired geometric optical path difference, as there is n_(o)physical curvature in this part of the lens, and for a birefringent lenswith thickness >1 mm and Δn approximately 0.1 it is far greater than the10 μm coherence length and is thus sufficient to prevent interferencefrom occurring. A correction similar to the GRIN method is made here, inwhich a compensating birefringent optical flat of thickness equal to thecenter thickness of the birefringent lens and cut with the sameorientation of its crystal axes is placed in the optical train with itsextraordinary axis rotated by 90° in the transverse plane relative tothe extraordinary axis of the birefringent lens (e.g., as shown in FIG.13). The wave that projects along the ordinary axis in the birefringentlens projects along the extraordinary axis of the compensatingbirefringent flat, and vice versa, so the non-spherical ΔOPD from thebirefringent lens is canceled by the compensating birefringent flat.

By using a BRF of the same thickness and cutting angle as the BRL, butrotated by 90 degrees in the plane orthogonal to the direction of EMpropagation, the OPD_(o) may be corrected without changing the relativedifference in the spherical curvatures of the two waves. The wave thatprojects along the ordinary axis in the BRL projects along theextraordinary axis of the BRF, and vice versa, so the non-sphericalOPD_(o) from the BRL is canceled by the BRF. Tilting the BRF slightlychanges the magnitude of this OPD matching effect to achieve maximuminterference contrast.

Another system shown in FIG. 12 incorporates only a BRF 1200 along witha glass lens 100 to effect the separation of the received wave from theobject 101 into two waves. Waves with positive spherical curvatureentering a medium experience a delay in achieving their focal point.This delay Δ is proportional to the thickness t and refractive index nof the medium:

$\begin{matrix}{\Delta = {t( {1 - \frac{1}{n}} )}} & (11)\end{matrix}$

It can readily be seen in the magnified part 1201 of FIG. 12 that a BRFcan delay the wave 1202 parallel to the ordinary axis and the wave 1203parallel to the extraordinary axis by different amounts due to thediffering refractive indices, which separates the focal planes 1204 and1205 of the two waves and allows for holography interference 305 to takeplace.

Some example embodiments use thin birefringent lenses in conjunctionwith classical refractive lenses in order to achieve a compoundbirefringent lens system (CBLS) that splits the received electromagneticradiation into two differentially phase-modulated components parallel tothe extraordinary and ordinary axes of the birefringent lens, thatpropagate along the optical axis. A “thin birefringent lens”, as used inthis disclosure, is a birefringent lens having a thickness (e.g., in thethickest section) that is less than or equal to 15% of its diameter. Insome embodiments, the thin birefringent lenses have a thickness that is10% or less than the diameter. Thin birefringent lenses having athickness that is 15% or less of the diameter are used as a closeapproximation of an idealized thin lens. In light of the fact thatbirefringent lenses made from birefringent single crystals may bedifficult and expensive to produce, it is notable that the deficienciesof other BRL types may be attenuated by judicious combination withclassical lenses. In this way, it may be considered that the bulk of thefocal power originates in the classical component of a CBLS, while thebirefringent component contributes just enough differential phasemodulation (e.g., approximately 5%; a 5% difference contributesapproximately 3-10% differential phase modulation) to produce thehologram interference with minimal amounts of overall aberration.

Birefringent components that are applicable to this concept includebirefringent Fresnel lenses made with either solid or liquid crystallinematerial, other optical elements made with patterned birefringent solidor liquid crystalline material, and micro- or nano-structuredmetamaterial optical elements; all of which will be referred to hereinas thin birefringent components (TBCs). Micro- or nano-structure opticalelements can include structures made of patterned silicon dioxide orother materials in which the patterns consist of nano-structures withdefined periodic radii, shapes and/or orientations that combine toproduce a focusing effect. Arbabi, A. et al. Subwavelength-thick lenseswith high numerical apertures and large efficiency based onhigh-contrast transmit arrays, Nat. Commun. 6:7069 doi:10.1038/ncomms8069 (2015), which is incorporated herein in its entirety,describes micro- and nano-structures. The notable potential advantagesof TBCs include (1) very low (e.g., 0 or substantially 0) overall phaseshift OPD_(o) of the sort described earlier in equation 10, (2) very low(e.g., 0 or substantially 0) spherical aberration due to their nearplanar structure and (3) the opportunity to encode other phase patternsbesides spherical quadratic patterns into the TBC for the purposes ofoptimizing the system for a given use or to correct for aberrations fromother components in the system.

Potential disadvantages of TBC's arise from their natures as diffractivelenses. Lenses made from TBCs (e.g., Fresnel lenses, lenses with micro-or nano-structures) generally have large chromatic shifts of focallength, which would have the undesirable effect of spreading the optimalhologram plane z_(h) over a large area of three-dimensional space in asystem with any wavelength bandwidth; and TBC-lenses also impart phaseaberrations such as diffraction rings and higher-order diffractioncomponents to transmitted beams. However, in the limit of TBC-lenseswith long focal lengths, these disadvantages may be mostly or entirelynegated for the purposes of FINCH or other holography by combining themwith classical lenses in CBLSs.

The chromatic variation in focal length for diffractive lenses isgenerally approximated as

$\begin{matrix}{\frac{\Delta \; f}{f} = \frac{\Delta\lambda}{\lambda}} & (12)\end{matrix}$

where f and λ are focal length and wavelength, respectively. However theAbbe number for diffractive lenses is −3.45, in distinction to thoserefractive lenses for which it is positive and of larger magnitude.Thus, while a TBC of 300 mm nominal focal length will have a focallength spread out over about 20 mm along the optical axis for a standard40 nm microscope bandwidth, for example, a TBC with a focal length ofseveral thousand mm (e.g., 5000 mm or approximately 5000 mm) can becoupled with a 300 mm (or approximately 300 mm) focal length classicallens to achieve a CBLS with much lower chromatic dispersion. Thisrelationship follows from the achromatic lens formula in equation 13a(of the sum to be minimized to achieve achromatic correction in atwo-lens system) and its logical consequence in equation 13b (for thevalue of the focal length f₂ that achieves best achromatic correctionfor a given pair of lenses):

$\begin{matrix}{\min ( {{f_{1}v_{d\; 1}} + {f_{2}v_{d\; 2}}} )} & ( {13a} ) \\{f_{1} = {- ( \frac{f_{1}v_{d\; 1}}{v_{d\; 2}} )}} & ( {13b} )\end{matrix}$

in which v_(d) is the Abbe number. The tables below show example systemsthat compare a single diffractive lens to a CBLS system that combined along focal length (e.g., 5000 mm or approximately 5000 mm) diffractivelens with a short focal length (e.g., 300 mm or approximately 300 mm)refractive lens. The chromatic shift in total focal length is much lowerfor the CBLS system, which will enable much better holographicperformance.

TABLE 2 chromatic dispersion of focal length of a diffractive lensDiffractive Diffractive λ (nm) λ (nm) lens nominal f lens actual factual nominal Δλ (nm) (mm) Δf (mm) (mm) 570 590 20 300 10.17 310.17 580590 10 300 5.08 305.08 590 590 0 300 0.00 300.00 600 590 −10 300 −5.08294.92 610 590 −20 300 −10.17 289.83 Legend: λ and f are lightwavelength and lens focal length, respectively. Δλ is the differencebetween the actual wavelength and the nominal wavelength for which thediffractive lens is designed for. Δf the change in diffractive lensfocal length resulting from the wavelength change. Diffractive lensactual f is the actual focal length at the specified actual wavelength.

TABLE 3 combined focal lengths of diffractive lens and classical lensDiffractive Diffractive classical lens Actual λ (nm) λ (nm) lens nominalf lens actual f approximate combined actual nominal Δλ (nm) (mm) Δf (mm)(mm) f (mm) f (mm) 570 590 20 5000 169.49 5169.49 300 283.55 580 590 105000 84.75 5084.75 300 283.29 590 590 0 5000 0.00 5000.00 300 283.02 600590 −10 5000 −84.75 4915.25 300 282.74 610 590 −20 5000 −169.49 4830.51300 282.46 Legend: λ and f are light wavelength and lens focal length,respectively. Δλ is the difference between the actual wavelength and thenominal wavelength for which the diffractive lens is designed for. Δf isthe change in diffractive lens focal length resulting from thewavelength change. Diffractive lens actual f is the actual focal lengthat the specified actual wavelength. Actual combined f is the combinedfocal length of the classical and diffractive lens calculated by thethin lens approximation and assuming no distance between the lenses.

From the above tables and equations, it can readily be seen thatcombining a classical lens with a TBC lens possessing one or twopolarization-dependent focal lengths can result in a CBLS with the twodifferentially focused or phase modulated electromagnetic beamsnecessary for FINCH or other holography, with relatively little (e.g.less than 2 mm) chromatic dispersion of the focal planes of each beam,and therefore with hologram distance z_(h) that is sharply defined andallows for high fringe contrast in the interference of the beams. It isalso noted that following equations 7 and 8, a CBLS designed on theseprinciples will also have significant potential flexibility in choice ofspacing factor s and hologram distance z_(h).

Furthermore, the diffractive aberrations introduced by TBCs derive fromthe sharp phase-transition regions or discontinuities in the component'sphase profile, such as the phase wrapping points of a Fresnel or otherTBC lens. With fewer phase wrapping regions, then, the number of phaseaberrations should be reduced. Since the number of phase wrappingregions is directly proportional to the focal length of a TBC lens,there will be very few phase wrapping regions in the limit of long focallength, and correspondingly fewer aberrations introduced. In the verylong focal length limit (e.g., in the limiting case where the focallength of the lens requires less than one wave of phase shift betweenthe center and edge of the lens, n_(o) phase wrapping regions occur),there might be n_(o) phase wrapping regions at all, and the system mighthe treated as a fully refractive one.

FIG. 13 shows various arrangements of optics in a birefringent lensincoherent interferometer according to some example embodiments. In (a)1301, the combination of a classical lens (CL) 1302 (in this caseconvergent) with a birefringent lens (BL) 1303 (in this case negative)to produce combined focal planes and hologram plane as in the text inequations 7 and 8b. Note that other combinations are possible as well.In (b) 1310, the combination of a birefringent lens (BL) 1311 with abirefringent compensating flat (CBF) 1312 to reduce the overall opticalpath difference encountered by the light propagating through theinterferometer is shown. The orientations of the extraordinary axes ofthe birefringent lens and the compensating flat are indicated by thearrow 1313 and target 1314, representing in the plane of the paper andorthogonal to it, respectively.

FIG. 14 schematically illustrates a FINCH microscope 1400 according tosome example embodiments. In (a) 1401, a standard fluorescent microscopearrangement in which fluorescent light emitted from a sample 1402 in themicroscope is shown. The fluorescent light emitted from the samplepasses through an infinity corrected objective 1403, after which pointit is split by a beam splitter 1404 into two polarized beams. The spolarization is directed through a microscope tube lens 1405 and theimage is captured on the widefield camera 1406 as in a classicalmicroscope. In (b) 1411, beam splitting into two orthogonally polarizedbeams typical of a FINCH hologram forming system configured with SLMbased or GRIN lens based or a birefringent crystal lens basedinterferometer is shown. The emitted light propagates through theobjective and a polarizing beamsplitter to an optical train that appliesdifferent spherical phases (focusing power) to different polarizationcomponents of the light beam, creating a pair of co-propagatingdifferentially focused beams with focal lengths fd1 and fd2. The beamspropagate until their interference is recorded at the ideal hologramplane located at distance zh. Following the recording of a set ofholograms used to recreate the complex field at the recording plane, afinal processed image is calculated by Fresnel propagation and asubsequent deconvolution. The p polarization is directed throughholographic optical elements 1416 which create holograms that arecaptured on the FINCH camera 1412. The phase of the hologram can bechanged by an optional polarization sensitive variable waveplate 1413 ifthe phase shifting holographic method is used. Additional contrast canalso be obtained by inclusion of an optional output polarizer 1414. Notshown for simplicity is a 4F relay system between (a) 1401 and (b) 1411.

Uniaxial birefringent α-BBO and calcite crystal materials were used inexample embodiments to create lens based in line incoherentinterferometers. These common path incoherent interferometers allowedthe inventors to make for the first time a FINCH holographic superresolution microscope with high magnification/numerical apertureobjectives. Birefringent crystal lens incoherent interferometers utilizenon quantized refractive lenses that create higher quality FINCHholograms because they are free of quantization errors and aberrationsinherent in SLM or GRIN lens devices used to produce FINCH holograms. Asimple fluorescence microscope incorporating these new birefringent lensinterferometers has a lateral point spread function (PSF) width of 149nm at 590 nm center wavelength with a 60×1.49 NA objective. This is asignificant improvement beyond the resolution of standard widefieldfluorescence microscopes and experimentally achieves sub diffractionsuper resolution performance predicted for FINCH fluorescencemicroscopy. Birefringent incoherent crystal interferometers arecontemplated in embodiments to aid other holographic applications.

Another use for a birefringent lens common path interferometer based onthese design principles is in the creation of the excitation beam inoptical scanning holography (OSH) and particularly in scanningholographic microscopy [J, Opt. Soc. Am. A 22, 892-898 (2005)]. Theexcitation beam in OSH microscopy is created by interfering two beamsthat are coherent with each other at the back focal plane of anobjective lens, resulting in the formation of an interferogram that isidentical to a Fresnel complex hologram. This excitation interferogramis then focused into the sample to produce a small excitation spot.Since the process of forming the excitation interferogram is identicalin principle to the formation of a FINCH hologram, it is clear thatcurrent methods for forming the excitation Hologram suffers from thesame drawbacks as many other hologram methods that FINCH was designed toremedy. Therefore a common-path birefringent interferometer shouldprovide the same advantages to the excitation interferogram in OSH as inFINCH, including ease and stability of alignment, and elimination ofsensitivity to environmental vibrations. Furthermore, given that bothOSH microscopy [J. OpL Soc. Am. A 22, 892-898 (2005)] and FINCH (asnoted above) are independently capable of super-resolution by factors ofup to 2 when compared to classical imaging methods, it is possible tocombine scanning OSH excitation with FINCH imaging detection to achieveeven further increases in super-resolution, potentially up to a factorof 4 compared to classical imaging. Additionally, it may be possible touse the same birefringent interferometer to produce both the excitationinterferogram and the emission FINCH hologram, simplifying andstabilizing a joint OSH/FINCH system even further.

Numerous modifications and variations of the present invention arepossible in light of the above teachings. It is therefore to beunderstood that within the scope of the appended claims, the inventionmay be practiced otherwise than as specifically described herein.

What is claimed is:
 1. An optical apparatus, comprising: a plurality oflenses including at least one thin birefringent lens, wherein theplurality of lenses are configured to: receive electromagnetic radiationfrom an object, wherein the electromagnetic radiation is incoherentlight; transform, by transmission using the at least one thinbirefringent lens, the received electromagnetic radiation to generatetwo or more differentially modulated electromagnetic waves propagatingin a common path; and provide for the differentially modulatedelectromagnetic waves to create electromagnetic interference.